- •Preface
- •Acknowledgements
- •Contents
- •1.1 Introduction
- •1.2 Classical Physics Between the End of the XIX and the Dawn of the XX Century
- •1.2.1 Maxwell Equations
- •1.2.2 Luminiferous Aether and the Michelson Morley Experiment
- •1.2.3 Maxwell Equations and Lorentz Transformations
- •1.3 The Principle of Special Relativity
- •1.3.1 Minkowski Space
- •1.4 Mathematical Definition of the Lorentz Group
- •1.4.1 The Lorentz Lie Algebra and Its Generators
- •1.4.2 Retrieving Special Lorentz Transformations
- •1.5 Representations of the Lorentz Group
- •1.5.1 The Fundamental Spinor Representation
- •1.6 Lorentz Covariant Field Theories and the Little Group
- •1.8 Criticism of Special Relativity: Opening the Road to General Relativity
- •References
- •2.1 Introduction
- •2.2 Differentiable Manifolds
- •2.2.1 Homeomorphisms and the Definition of Manifolds
- •2.2.2 Functions on Manifolds
- •2.2.3 Germs of Smooth Functions
- •2.3 Tangent and Cotangent Spaces
- •2.4 Fibre Bundles
- •2.5 Tangent and Cotangent Bundles
- •2.5.1 Sections of a Bundle
- •2.5.2 The Lie Algebra of Vector Fields
- •2.5.3 The Cotangent Bundle and Differential Forms
- •2.5.4 Differential k-Forms
- •2.5.4.1 Exterior Forms
- •2.5.4.2 Exterior Differential Forms
- •2.6 Homotopy, Homology and Cohomology
- •2.6.1 Homotopy
- •2.6.2 Homology
- •2.6.3 Homology and Cohomology Groups: General Construction
- •2.6.4 Relation Between Homotopy and Homology
- •References
- •3.1 Introduction
- •3.2 A Historical Outline
- •3.2.1 Gauss Introduces Intrinsic Geometry and Curvilinear Coordinates
- •3.2.3 Parallel Transport and Connections
- •3.2.4 The Metric Connection and Tensor Calculus from Christoffel to Einstein, via Ricci and Levi Civita
- •3.2.5 Mobiles Frames from Frenet and Serret to Cartan
- •3.3 Connections on Principal Bundles: The Mathematical Definition
- •3.3.1 Mathematical Preliminaries on Lie Groups
- •3.3.1.1 Left-/Right-Invariant Vector Fields
- •3.3.1.2 Maurer-Cartan Forms on Lie Group Manifolds
- •3.3.1.3 Maurer Cartan Equations
- •3.3.2 Ehresmann Connections on a Principle Fibre Bundle
- •3.3.2.1 The Connection One-Form
- •Gauge Transformations
- •Horizontal Vector Fields and Covariant Derivatives
- •3.4 Connections on a Vector Bundle
- •3.5 An Illustrative Example of Fibre-Bundle and Connection
- •3.5.1 The Magnetic Monopole and the Hopf Fibration of S3
- •The U(1)-Connection of the Dirac Magnetic Monopole
- •3.6.1 Signatures
- •3.7 The Levi Civita Connection
- •3.7.1 Affine Connections
- •3.7.2 Curvature and Torsion of an Affine Connection
- •Torsion and Torsionless Connections
- •The Levi Civita Metric Connection
- •3.8 Geodesics
- •3.9 Geodesics in Lorentzian and Riemannian Manifolds: Two Simple Examples
- •3.9.1 The Lorentzian Example of dS2
- •3.9.1.1 Null Geodesics
- •3.9.1.2 Time-Like Geodesics
- •3.9.1.3 Space-Like Geodesics
- •References
- •4.1 Introduction
- •4.2 Keplerian Motions in Newtonian Mechanics
- •4.3 The Orbit Equations of a Massive Particle in Schwarzschild Geometry
- •4.3.1 Extrema of the Effective Potential and Circular Orbits
- •Minimum and Maximum
- •Energy of a Particle in a Circular Orbit
- •4.4 The Periastron Advance of Planets or Stars
- •4.4.1 Perturbative Treatment of the Periastron Advance
- •References
- •5.1 Introduction
- •5.2 Locally Inertial Frames and the Vielbein Formalism
- •5.2.1 The Vielbein
- •5.2.2 The Spin-Connection
- •5.2.3 The Poincaré Bundle
- •5.3 The Structure of Classical Electrodynamics and Yang-Mills Theories
- •5.3.1 Hodge Duality
- •5.3.2 Geometrical Rewriting of the Gauge Action
- •5.3.3 Yang-Mills Theory in Vielbein Formalism
- •5.4 Soldering of the Lorentz Bundle to the Tangent Bundle
- •5.4.1 Gravitational Coupling of Spinorial Fields
- •5.5 Einstein Field Equations
- •5.6 The Action of Gravity
- •5.6.1 Torsion Equation
- •5.6.1.1 Torsionful Connections
- •The Torsion of Dirac Fields
- •Dilaton Torsion
- •5.6.2 The Einstein Equation
- •5.6.4 Examples of Stress-Energy-Tensors
- •The Stress-Energy Tensor of the Yang-Mills Field
- •The Stress-Energy Tensor of a Scalar Field
- •5.7 Weak Field Limit of Einstein Equations
- •5.7.1 Gauge Fixing
- •5.7.2 The Spin of the Graviton
- •5.8 The Bottom-Up Approach, or Gravity à la Feynmann
- •5.9 Retrieving the Schwarzschild Metric from Einstein Equations
- •References
- •6.1 Introduction and Historical Outline
- •6.2 The Stress Energy Tensor of a Perfect Fluid
- •6.3 Interior Solutions and the Stellar Equilibrium Equation
- •6.3.1 Integration of the Pressure Equation in the Case of Uniform Density
- •6.3.1.1 Solution in the Newtonian Case
- •6.3.1.2 Integration of the Relativistic Pressure Equation
- •6.3.2 The Central Pressure of a Relativistic Star
- •6.4 The Chandrasekhar Mass-Limit
- •6.4.1.1 Idealized Models of White Dwarfs and Neutron Stars
- •White Dwarfs
- •Neutron Stars
- •6.4.2 The Equilibrium Equation
- •6.4.3 Polytropes and the Chandrasekhar Mass
- •6.5 Conclusive Remarks on Stellar Equilibrium
- •References
- •7.1 Introduction
- •7.1.1 The Idea of GW Detectors
- •7.1.2 The Arecibo Radio Telescope
- •7.1.2.1 Discovery of the Crab Pulsar
- •7.1.2.2 The 1974 Discovery of the Binary System PSR1913+16
- •7.1.3 The Coalescence of Binaries and the Interferometer Detectors
- •7.2 Green Functions
- •7.2.1 The Laplace Operator and Potential Theory
- •7.2.2 The Relativistic Propagators
- •7.2.2.1 The Retarded Potential
- •7.3 Emission of Gravitational Waves
- •7.3.1 The Stress Energy 3-Form of the Gravitational Field
- •7.3.2 Energy and Momentum of a Plane Gravitational Wave
- •7.3.2.1 Calculation of the Spin Connection
- •7.3.3 Multipolar Expansion of the Perturbation
- •7.3.3.1 Multipolar Expansion
- •7.3.4 Energy Loss by Quadrupole Radiation
- •7.3.4.1 Integration on Solid Angles
- •7.4 Quadruple Radiation from the Binary Pulsar System
- •7.4.1 Keplerian Parameters of a Binary Star System
- •7.4.2 Shrinking of the Orbit and Gravitational Waves
- •7.4.2.1 Calculation of the Moment of Inertia Tensor
- •7.4.3 The Fate of the Binary System
- •7.4.4 The Double Pulsar
- •7.5 Conclusive Remarks on Gravitational Waves
- •References
- •Appendix A: Spinors and Gamma Matrix Algebra
- •A.2 The Clifford Algebra
- •A.2.1 Even Dimensions
- •A.2.2 Odd Dimensions
- •A.3 The Charge Conjugation Matrix
- •A.4 Majorana, Weyl and Majorana-Weyl Spinors
- •Appendix B: Mathematica Packages
- •B.1 Periastropack
- •Programme
- •Main Programme Periastro
- •Subroutine Perihelkep
- •Subroutine Perihelgr
- •Examples
- •B.2 Metrigravpack
- •Metric Gravity
- •Routines: Metrigrav
- •Mainmetric
- •Metricresume
- •Routine Metrigrav
- •Calculation of the Ricci Tensor of the Reissner Nordstrom Metric Using Metrigrav
- •Index
Acknowledgements
With great pleasure I would like to thank my collaborators and colleagues Pietro Antonio Grassi, Igor Pesando and Mario Trigiante for the many suggestions and discussions we had during the writing of the present book and also for their critical reading of several chapters. Similarly I express my gratitude to the Editors of Springer-Verlag, in particular to Dr. Maria Bellantone, for their continuous assistance, constructive criticism and suggestions.
My thoughts, while finishing the writing of these volumes, that occurred during solitary winter week-ends in Moscow, were frequently directed to my late parents, whom I miss very much and I will never forget. To them I also express my gratitude for all what they taught me in their life, in particular to my father who, with his own example, introduced me, since my childhood, to the great satisfaction and deep suffering of writing books.
Furthermore it is my pleasure to thank my very close friend and collaborator Aleksander Sorin for his continuous encouragement and for many precious consultations.
xiii
Contents
1 Special Relativity: Setting the Stage . . . . . . . . . . . . . . . . . . . |
1 |
||
1.1 |
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
1 |
|
1.2 |
Classical Physics Between the End of the XIX and the Dawn |
|
|
|
of the XX Century . . . . . . . . . . . . . . . . . . . . . . . . . . |
2 |
|
|
1.2.1 |
Maxwell Equations . . . . . . . . . . . . . . . . . . . . . |
2 |
|
1.2.2 Luminiferous Aether and the Michelson Morley Experiment |
4 |
|
|
1.2.3 Maxwell Equations and Lorentz Transformations . . . . . . |
6 |
|
1.3 |
The Principle of Special Relativity . . . . . . . . . . . . . . . . . |
8 |
|
|
1.3.1 |
Minkowski Space . . . . . . . . . . . . . . . . . . . . . . |
10 |
1.4 |
Mathematical Definition of the Lorentz Group . . . . . . . . . . . |
15 |
|
|
1.4.1 The Lorentz Lie Algebra and Its Generators . . . . . . . . |
16 |
|
|
1.4.2 Retrieving Special Lorentz Transformations . . . . . . . . |
18 |
|
1.5 |
Representations of the Lorentz Group . . . . . . . . . . . . . . . . |
19 |
|
|
1.5.1 The Fundamental Spinor Representation . . . . . . . . . . |
20 |
|
1.5.2The Two-Valued Homomorphism SO(1, 3) SL(2, C)
in the Four-Dimensional Case . . . . . . . . . . . . . . . . 22 1.6 Lorentz Covariant Field Theories and the Little Group . . . . . . . 23 1.6.1 Representations of the Massless Little Group in D = 4 . . . 27
1.7 Noether’s Theorem, Noether’s Currents and the Stress-Energy
Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.8 Criticism of Special Relativity: Opening the Road to General
Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Basic Concepts About Manifolds and Fibre Bundles . . . . . . . . . |
35 |
|
2.1 |
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
35 |
2.2 |
Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . |
36 |
|
2.2.1 Homeomorphisms and the Definition of Manifolds . . . . . |
37 |
|
2.2.2 Functions on Manifolds . . . . . . . . . . . . . . . . . . . |
42 |
|
2.2.3 Germs of Smooth Functions . . . . . . . . . . . . . . . . . |
43 |
2.3 |
Tangent and Cotangent Spaces . . . . . . . . . . . . . . . . . . . |
44 |
xv
xvi |
|
|
|
Contents |
|
|
2.3.1 Tangent Vectors in a Point p M . . . . . |
. . . . . . . . |
45 |
||
|
2.3.2 Differential Forms in a Point p M . . . . |
. . . . . . . . |
49 |
||
2.4 |
Fibre Bundles . . . . . . . . . |
. . . . . . . . . . . |
. . . . . . . . |
51 |
|
2.5 |
Tangent and Cotangent Bundles |
. . . . . . . . . . . |
. . . . . . . . |
58 |
|
|
2.5.1 |
Sections of a Bundle . . |
. . . . . . . . . . . |
. . . . . . . . |
60 |
|
2.5.2 The Lie Algebra of Vector Fields . . . . . . . |
. . . . . . . |
62 |
||
|
2.5.3 The Cotangent Bundle and Differential Forms |
. . . . . . . |
64 |
||
|
2.5.4 |
Differential k-Forms . . . |
. . . . . . . . . . . |
. . . . . . . |
66 |
2.6 |
Homotopy, Homology and Cohomology . . . . . . . |
. . . . . . . |
70 |
||
|
2.6.1 |
Homotopy . . . . . . . . |
. . . . . . . . . . . |
. . . . . . . |
72 |
|
2.6.2 |
Homology . . . . . . . . |
. . . . . . . . . . . |
. . . . . . . |
75 |
|
2.6.3 Homology and Cohomology Groups: General Construction |
81 |
|||
|
2.6.4 Relation Between Homotopy and Homology . |
. . . . . . . |
83 |
||
References . |
. . . . . . . . . . . . . . |
. . . . . . . . . . . |
. . . . . . . |
84 |
|
3 Connections and Metrics . . . . . . . |
. . . . . . . . . . . |
. . . . . . . |
85 |
||
3.1 |
Introduction . . . . . . . . . . . |
. . . . . . . . . . . |
. . . . . . . |
85 |
|
3.2 |
A Historical Outline . . . . . . . |
. . . . . . . . . . . |
. . . . . . . |
86 |
|
3.2.1Gauss Introduces Intrinsic Geometry and Curvilinear
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . |
87 |
3.2.2Bernhard Riemann Introduces n-Dimensional Metric
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . |
91 |
3.2.3 Parallel Transport and Connections . . . . . . . . . . . . . |
94 |
3.2.4The Metric Connection and Tensor Calculus
from Christoffel to Einstein, via Ricci and Levi Civita . . . 94
|
3.2.5 Mobiles Frames from Frenet and Serret to Cartan . . . |
. . |
102 |
3.3 |
Connections on Principal Bundles: The Mathematical Definition |
. |
108 |
|
3.3.1 Mathematical Preliminaries on Lie Groups . . . . . . . |
. . |
108 |
|
3.3.2 Ehresmann Connections on a Principle Fibre Bundle . . |
. . |
118 |
3.4 |
Connections on a Vector Bundle . . . . . . . . . . . . . . . . . |
. . |
127 |
3.5 |
An Illustrative Example of Fibre-Bundle and Connection . . . |
. . |
130 |
3.5.1 The Magnetic Monopole and the Hopf Fibration of S3 . . . 130
3.6Riemannian and Pseudo-Riemannian Metrics: The Mathematical
|
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
136 |
|
|
3.6.1 |
Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . |
137 |
3.7 |
The Levi Civita Connection . . . . . . . . . . . . . . . . . . . . . |
139 |
|
|
3.7.1 |
Affine Connections . . . . . . . . . . . . . . . . . . . . . |
140 |
|
3.7.2 Curvature and Torsion of an Affine Connection . . . . . . . |
141 |
|
3.8 |
Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
144 |
|
3.9 |
Geodesics in Lorentzian and Riemannian Manifolds: Two Simple |
|
|
|
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
145 |
|
|
3.9.1 The Lorentzian Example of dS2 . . . . . . . . . . . . . . . |
146 |
|
3.9.2The Riemannian Example of the Lobachevskij-Poincaré
Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
151 |
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
154 |
Contents |
|
|
xvii |
4 Motion of a Test Particle in the Schwarzschild Metric . . . . . . . . . |
157 |
||
4.1 |
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
157 |
|
4.2 |
Keplerian Motions in Newtonian Mechanics . . . . . . . . . . . . |
160 |
|
4.3 |
The Orbit Equations of a Massive Particle in Schwarzschild |
|
|
|
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
162 |
|
|
4.3.1 |
Extrema of the Effective Potential and Circular Orbits . . . |
165 |
4.4 |
The Periastron Advance of Planets or Stars . . . . . . . . . . . . . |
170 |
|
|
4.4.1 |
Perturbative Treatment of the Periastron Advance . . . . . |
174 |
4.5 |
Light-Like Geodesics in the Schwarzschild Metric |
|
|
|
and the Deflection of Light Rays . . . . . . . . . . . . . . . . . . |
179 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
185 |
|
5 Einstein Versus Yang-Mills Field Equations: The Spin Two |
|
||
Graviton and the Spin One Gauge Bosons . . . . . . . . . . . . . . . |
187 |
||
5.1 |
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
187 |
|
5.2 |
Locally Inertial Frames and the Vielbein Formalism . . . . . . . . |
189 |
|
|
5.2.1 |
The Vielbein . . . . . . . . . . . . . . . . . . . . . . . . . |
192 |
|
5.2.2 |
The Spin-Connection . . . . . . . . . . . . . . . . . . . . |
193 |
|
5.2.3 |
The Poincaré Bundle . . . . . . . . . . . . . . . . . . . . . |
194 |
5.3 |
The Structure of Classical Electrodynamics and Yang-Mills |
|
|
|
Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
195 |
|
|
5.3.1 |
Hodge Duality . . . . . . . . . . . . . . . . . . . . . . . . |
198 |
|
5.3.2 Geometrical Rewriting of the Gauge Action . . . . . . . . |
199 |
|
|
5.3.3 Yang-Mills Theory in Vielbein Formalism . . . . . . . . . |
200 |
|
5.4 |
Soldering of the Lorentz Bundle to the Tangent Bundle . . . . . . |
204 |
|
|
5.4.1 Gravitational Coupling of Spinorial Fields . . . . . . . . . |
207 |
|
5.5 |
Einstein Field Equations . . . . . . . . . . . . . . . . . . . . . . . |
209 |
|
5.6 |
The Action of Gravity . . . . . . . . . . . . . . . . . . . . . . . . |
211 |
|
|
5.6.1 |
Torsion Equation . . . . . . . . . . . . . . . . . . . . . . . |
214 |
|
5.6.2 |
The Einstein Equation . . . . . . . . . . . . . . . . . . . . |
217 |
5.6.3Conservation of the Stress-Energy Tensor and Symmetries
|
|
of the Gravitational Action . . . . . . . . . . . . . . |
. . . 218 |
|
5.6.4 |
Examples of Stress-Energy-Tensors . . . . . . . . . . . |
. . 219 |
5.7 |
Weak Field Limit of Einstein Equations . . . . . . . . . . . . . |
. . 220 |
|
|
5.7.1 |
Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . |
. . 222 |
|
5.7.2 The Spin of the Graviton . . . . . . . . . . . . . . . . |
. . 225 |
|
5.8 |
The Bottom-Up Approach, or Gravity à la Feynmann . . . . . |
. . 227 |
|
5.9 |
Retrieving the Schwarzschild Metric from Einstein Equations |
. . . 233 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . 236 |
|
6 Stellar Equilibrium: Newton’s Theory, General Relativity, |
|
||
Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . |
. . 237 |
||
6.1 |
Introduction and Historical Outline . . . . . . . . . . . . . . . |
. . 237 |
|
6.2 |
The Stress Energy Tensor of a Perfect Fluid . . . . . . . . . . . |
. . 242 |
|
6.3 |
Interior Solutions and the Stellar Equilibrium Equation . . . . |
. . 245 |
|
xviii |
Contents |
6.3.1Integration of the Pressure Equation in the Case
of Uniform Density . . . . . . . . . . . . . . . . . . . . . |
250 |
6.3.2 The Central Pressure of a Relativistic Star . . . . . . . . . |
254 |
6.4 The Chandrasekhar Mass-Limit . . . . . . . . . . . . . . . . . . . |
256 |
6.4.1The Degenerate Fermi Gas of Very Many Spin One-Half
|
Particles . . . . . . . . . . . . . . . . . |
. . . . . . . . . . 256 |
6.4.2 |
The Equilibrium Equation . . . . . . . . |
. . . . . . . . . . 264 |
6.4.3 |
Polytropes and the Chandrasekhar Mass |
. . . . . . . . . . 267 |
6.5 Conclusive Remarks on Stellar Equilibrium . . . |
. . . . . . . . . . 270 |
|
References . . . . . . . . . . . . . . . . . . . . . . . |
. . . . . . . . . . 271 |
|
7 Gravitational Waves and the Binary Pulsars . . . . . . . . . . . . . . 273 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7.1.1 The Idea of GW Detectors . . . . . . . . . . . . . . . . . . 274 7.1.2 The Arecibo Radio Telescope . . . . . . . . . . . . . . . . 276
7.1.3The Coalescence of Binaries and the Interferometer
|
|
Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . |
278 |
7.2 |
Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . |
280 |
|
|
7.2.1 |
The Laplace Operator and Potential Theory . . . . . . . . . |
283 |
|
7.2.2 |
The Relativistic Propagators . . . . . . . . . . . . . . . . . |
284 |
7.3 |
Emission of Gravitational Waves . . . . . . . . . . . . . . . . . . |
286 |
|
7.3.1The Stress Energy 3-Form of the Gravitational Field . . . . 286
7.3.2 Energy and Momentum of a Plane Gravitational Wave . . . 288
7.3.3Multipolar Expansion of the Perturbation . . . . . . . . . . 291
7.3.4 |
Energy Loss by Quadrupole Radiation . . . . . . . . . . . |
295 |
7.4 Quadruple Radiation from the Binary Pulsar System . . . . . . . . |
298 |
|
7.4.1 |
Keplerian Parameters of a Binary Star System . . . . . . . |
298 |
7.4.2 |
Shrinking of the Orbit and Gravitational Waves . . . . . . . |
301 |
7.4.3 |
The Fate of the Binary System . . . . . . . . . . . . . . . |
306 |
7.4.4 |
The Double Pulsar . . . . . . . . . . . . . . . . . . . . . . |
307 |
7.5 Conclusive Remarks on Gravitational Waves . . . . . . . . . . . . |
308 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
309 |
8 Conclusion of Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . |
311 |
|
Appendix A Spinors and Gamma Matrix Algebra . . . . . . . . . . . |
312 |
|
A.1 |
Introduction to the Spinor Representations of SO(1, D − 1) |
312 |
A.2 |
The Clifford Algebra . . . . . . . . . . . . . . . . . . . . |
312 |
A.3 |
The Charge Conjugation Matrix . . . . . . . . . . . . . . . |
314 |
A.4 |
Majorana, Weyl and Majorana-Weyl Spinors . . . . . . . . |
316 |
A.5 |
A Particularly Useful Basis for D = 4 γ -Matrices . . . . . |
317 |
Appendix B |
Mathematica Packages . . . . . . . . . . . . . . . . . . . |
318 |
B.1 |
Periastropack . . . . . . . . . . . . . . . . . . . . . . . . . |
318 |
B.2 |
Metrigravpack . . . . . . . . . . . . . . . . . . . . . . . . |
324 |
Index . . . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
331 |